Logical form of Either and Neither: Alice in room

Question

This is one of the problem I have been working:

Analyze the logical forms of the following statements:
(a) Either Alice or Bob is not in the room.
(a) Neither Alice nor Bob is in the room.

Now, I have come up with the following logical forms:

A = Alice is in the room.
B = Bob is in the room.

(a) (A ∨ ¬B) ∧ (¬A ∨ B)
(b) ¬(A ∧ B)

Now, I have been verifying my answer from here and it has an completely different answer:

(a) ¬A ∨ ¬B
(a) ¬A ∧ ¬B

Can somebody provide more insights into this ?

Answer 1

You might benefit from an answer that explains where you went wrong in the train of thought that led to your answers. Unfortunately, you don't describe your train of thought, and I can't think of a plausible one that would lead to your answers. So I'll just indicate what your answers are saying, without worrying about how you got them.

In part (a), your answer $(A\lor\neg B)\land(\neg A\lor B)$ is logically equivalent to $(A\iff B)$, i.e., "Alice is in the room if and only Bob is." In particular, it's true if both of them are in the room, and it's false if one is in the room and the other is out. (You should check what I wrote here, using truth tables.) Both of these disagree with what (a) is supposed to say.

In part (b), your answer $\neg(A\land B)$ says that it's not the case that they're both in the room; equivalently, at least one is out. So it would be true if one is in the room and the other is out. (Again, check using truth tables.) That disagrees with what (b) is supposed to say.

By the way, you should probably also work out the truth tables of the correct answers, to see that they do indeed say what they're supposed to say.

Answer 2

(a) "Either Alice or Bob is not in the room" translates to "(Alice is not in the room) OR (Bob is not in the room)". That is: (NOT A) OR (NOT B).

(b) "Neither Alice nor Bob is in the room" translates to "(Alice is not in the room) AND (Bob is not in the room)". That is: (NOT A) AND (NOT B).

Answer 3

A = Alice is in the room. B = Bob is in the room.

(a) (A ∨ ¬B) ∧ (¬A ∨ B)

(b) ¬(A ∧ B)

What does your (a) say? Either Alice is in the room and Bob isn't or (vice versa) Alice is isn't in the room and Bob is. In other words, (a) comes to Exactly one of Alice and Bob is in the room. But that isn't what Either Alice or Bob is not in the room strictly speaking says. After all it would be consistent to say Either Alice or Bob is not in the room, for a start (they can't bear being in the same room!), and maybe neither are. In other words, the disjunction here should by default (in the absence of additional information) be taken as inclusive, and you have assumed it is exclusive. That's not a horrible mistake, but it does seem to be a mistake about the literal context of the given disjunction as it stands.

What does your (b) say? It says that it isn't the case that Alice-and-Bob-are-in-the-room. I.e. it says They are not both in the room. Which is obviously quite different from saying that neither is in the room. So translating the 'neither' proposition by (b) is, I'm afraid, a rather horrible mistake!